If true, Targett would instantly bring competition to the Villa squad as they currently only have Wales international Neil Taylor at left-back.
Targett made 16 appearances for Saints in the Premier League last season.
Before that, the Saints academy product courted interest from Fulham following a successful loan spell at the Whites which helped them achieve promotion to the Premier League in 2017/18.
Fulham were promoted via the play-offs at Villa’s expense.
The Villains are back in the Premier League after beating Frank Lampard’s Derby County in the play-offs last season.
Meanwhile, Charlie Austin is apparently wanted by another Premier League newcomer in Sheffield United.
Sky Sports’ Transfer Centre yesterday reported the Blades are in search of a Premier League striker upon being promoted to the Premier League, and Austin allegedly fits the bill.
Southampton need to offload a number of players in this summer’s transfer window and it would appear manager Ralph Hasenhuttl wouldn’t be too upset at losing Austin - even if Saints’ attacking options are scarce at present.